Solution:

M.Tech Semester-II : Structural Dynamics-Tutorial 1Calculate the Natural Frequency for the following systems (1-2):

(1)EI = Constant(2)

(3) In the spring mass system, the natural frequency of vibration is 3.5Hz. When the amount of suspended mass is increased by 50 kg, the natural frequency is lowered to 2.9Hz. Find the original mass of the system and spring constant.(4) The stiffness and damping properties of a mass-spring-damper system are to be determined by a free vibration test. The mass is given as 0.18 N-sec2/cm. In this test the mass is displaced 2.54cm by a hydraulic jack and suddenly released. At the end of 20 complete cycles the time is 3sec and amplitude is 0.508cm. Determine the Stiffness and Damping coefficient.

(5) A vibration system consists of a mass 50 kg, a spring of stiffness 30 kN/m and a damper. The damping provided is only 20% of the critical value.

Determine, (a) the critical damping coefficient, (b) damping coefficient, (c) natural frequency of the damped vibrations, (d) the logarithmic decrement and (e) the ratio of two consecutive amplitudes.

(6) A single storey building is idealized as a rigid girder supported by weightless columns. In order to evaluate dynamic properties of structure, a free vibration test is conducted in which the rigid girder is displaced laterally by a hydraulic jack and suddenly released. During the jacking operation, it is observed that jacking force of 9072 kg is required to displace the girder by 0.508 cm. After the instantaneous release of this initial displacement, the maximum displacement on the return swing is only 0.406 cm and period of displacement cycle is 1.4 sec.

Determine (a) Eff. Wt. of girder (b) Undamped frequency of vibration

(c) Damping coefficient (d) Amplitude after 6 cycles. (7) A vertical single cylinder diesel engine of 500 kg is mounted on springs with k = 200 kN/m and dampers with = 0.2. The rotating parts are well balanced. The mass of the equivalent reciprocating parts is 10 kg and the stroke is 200mm. Find the dynamic amplitude of the vertical motion, the transmissibility and the force transmitted to the foundation, if the engine is operated at 200 rpm.

(8) An automobile is travelling along a multi span elevated roadway supported every 30m. Long term creep has resulted in 150mm deflection at middle of each span. The road profile can be approximated as sinusoidal with amplitude of 75mm and a period of 30m. A SDOF system is a simple idealization of an automobile appropriate for a first approximation study of the ride quality of the vehicle. When fully loaded, the weight of the automobile is 1818 kg. The effective stiffness of the automobile suspension is 140.44 N/mm. Its viscous damping coefficient is such that the damping ratio is 40%.

Determine

(a) the amplitude of vertical motion when the automobile is travelling at 64 kmph

(b) speed of vehicle which could produce a resonance condition